Dynamical spectral rigidity and determination for billiard systems

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The goal of inverse problems is the reconstruction of an object from a set of coarse observations. Inverse problems constitute a surprisingly broad class of problems with far-reaching applications in virtually any field of science: high energy physics (identifying particles created in scattering events), medicine (CT scans), astrophysics (detecting elements in the photosphere of stars which are billions of light-years away from us), image processing (de-noising and de-blurring of digital images), Big Data and Machine Learning. In this proposal I will describe the analysis of an inverse problem that can be purely formulated in the context of (classical) dynamical systems. Consider the trajectories of a particle that moves freely inside a planar domain and is subject to elastic reflections upon collisions with the boundary of the domain. We call periodic those trajectories that repeat themselves after a finite amount of time: such trajectories trace closed polygons inscribed in the domain. We call Length Spectrum of the domain the set of perimeters of all such polygons. We can then formulate the following inverse dynamical problem: Dynamical Spectral Determination: is it possible to identify the domain (modulo isometries) by the knowledge of its Length Spectrum? The above question, e.g. in the very natural class of smooth domains, is still wide open, and it is unarguably considered an extremely challenging problem. Sarnak conjectured that smooth domains are locally determined by their Laplace spectrum (a question that -quoting M. Kac- is often phrased as: "Can one hear the shape of a drum?"). Due to the tight connection between the Laplace (quantum) and dynamical (classical) spectral problems established by the Wave Trace formula, we find natural to study this conjecture in the dynamical setting. As a first step, we may consider a deformational problem: we say that a domain is dynamically spectrally rigid if all smooth deformations of the domain that preserve its Length Spectrum are necessarily isometries. In the past few years, together with my collaborators, we proved dynamical spectral rigidity for symmetric convex billiards close enough to disks. Also, we proved dynamical spectral determination for a class of (symmetric) analytic open dispersing billiards (such are systems whose dynamics is reminiscent of geodesic flow on manifolds with negative curvature). In the next several years, my research team and I will leverage on the breakthrough techniques that have been developed for the above results to move towards Sarnak's conjecture. On the one hand I will set to prove (local) spectral determination results for smooth convex billiards (possibly with symmetries); on the other hand I will push my work on hyperbolic billiards towards the problem of (local) spectral determination in the smooth category. I believe that these results will be attainable in this decade, and the Discovery Grant will play a major role in their development.